Post on 20-Mar-2016
description
Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP1)
Observations:
Sky Maps, Systematic Errors, and Basic Results
N. Jarosik2, C. L. Bennett3, J. Dunkley4, B. Gold3, M. R. Greason5, M. Halpern6, R. S.
Hill5, G. Hinshaw7, A. Kogut7, E. Komatsu8, D. Larson3, M. Limon9, S. S. Meyer10, M. R.
Nolta11, N. Odegard5, L. Page2, K. M. Smith12, D. N. Spergel12,13, G. S. Tucker14, J. L.
Weiland5, E. Wollack7, E. L. Wright15
ABSTRACT
New full sky temperature and polarization maps based on seven years of
data from WMAP are presented. The new results are consistent with previous
results, but have improved due to reduced noise from the additional integration
time, improved knowledge of the instrument performance, and improved data
analysis procedures. The improvements are described in detail.
1WMAP is the result of a partnership between Princeton University and NASA’s Goddard Space Flight
Center. Scientific guidance is provided by the WMAP Science Team.
2Dept. of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544-0708
3Dept. of Physics & Astronomy, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD
21218-2686
4Astrophysics, University of Oxford, Keble Road, Oxford, OX1 3RH, UK
5ADNET Systems, Inc., 7515 Mission Dr., Suite A100 Lanham, Maryland 20706
6Dept. of Physics and Astronomy, University of British Columbia, Vancouver, BC Canada V6T 1Z1
7Code 665, NASA/Goddard Space Flight Center, Greenbelt, MD 20771
8Univ. of Texas, Austin, Dept. of Astronomy, 2511 Speedway, RLM 15.306, Austin, TX 78712
9Columbia Astrophysics Laboratory, 550 W. 120th St., Mail Code 5247, New York, NY 10027-6902
10Depts. of Astrophysics and Physics, KICP and EFI, University of Chicago, Chicago, IL 60637
11Canadian Institute for Theoretical Astrophysics, 60 St. George St, University of Toronto, Toronto, ON
Canada M5S 3H8
12Dept. of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544-1001
13Princeton Center for Theoretical Physics, Princeton University, Princeton, NJ 08544
14Dept. of Physics, Brown University, 182 Hope St., Providence, RI 02912-1843
15UCLA Physics & Astronomy, PO Box 951547, Los Angeles, CA 90095–1547
– 2 –
The seven year data set is well fit by a minimal six-parameter flat ΛCDM
model. The parameters for this model, using the WMAP data in conjunction
with baryon acoustic oscillation data from the Sloan Digital Sky Survey and priors
on H0 from Hubble Space Telescope observations, are: Ωbh2 = 0.02260±0.00053,
Ωch2 = 0.1123± 0.0035, ΩΛ = 0.728+0.015
−0.016, ns = 0.963± 0.012, τ = 0.087± 0.014
and σ8 = 0.809 ± 0.024 (68 % CL uncertainties).
The temperature power spectrum signal-to-noise ratio per multipole is greater
that unity for multipoles ℓ . 919, allowing a robust measurement of the third
acoustic peak. This measurement results in improved constraints on the matter
density, Ωmh2 = 0.1334+0.0056−0.0055, and the epoch of matter-radiation equality, zeq =
3196+134−133, using WMAP data alone.
The new WMAP data, when combined with smaller angular scale microwave
background anisotropy data, results in a 3σ detection of the abundance of pri-
mordial Helium, YHe = 0.326 ± 0.075. When combined with additional external
data sets, the WMAP data also yield better determinations of the total mass of
neutrinos,∑
mν =< 0.58 eV (95% CL), and the effective number of neutrino
species, Neff = 4.34+0.86−0.88. The power-law index of the primordial power spectrum
is now determined to be ns = 0.963 ± 0.012, excluding the Harrison-Zel’dovich-
Peebles spectrum by > 3σ
These new WMAP measurements provide important tests of Big Bang cos-
mology.
Subject headings: cosmic microwave background, space vehicles: instruments
1. Introduction
The Wilkinson Microwave Anisotropy Probe (WMAP ) is a NASA sponsored satellite
designed to map the Cosmic Microwave Background (CMB) radiation over the entire sky
in five frequency bands. It was launched in June 2001 from Kennedy Space Flight Center
and began surveying the sky from its orbit around the the Earth-Sun L2 point in August
2001. This work and the accompanying papers comprise the fourth in a series of biennial
data releases and incorporates seven years of observational data.
Results from the one-year, three-year and five-year results are summarized in Bennett
et al. (2003a), Jarosik et al. (2007) and Hinshaw et al. (2009) respectively, and references
therein. An overall description of the mission including instrument nomenclature is contained
in Bennett et al. (2003b) and Limon et al. (2010), while details of the optical system and
radiometers can be found in Page et al. (2003b) and Jarosik et al. (2003).
– 3 –
The primary data product of WMAP are sets of calibrated sky maps at five frequency
bands centered at 23 GHz (K band), 33 GHz (Ka band), 41 GHz (Q band), 61 GHz (V
band) and 94 GHz (W band), including measured noise levels and beam transfer functions
that describe the smoothing of the sky signal resulting from the beam geometries. These
maps are provided for Stokes I, Q and U parameters on a year by year basis and in a year
co-added format, and at several pixel resolutions appropriate for various analyses. Changes
relative to the previous data release include the inclusion of seven years of observational
data, a new masking procedure that simplifies the map-making process, and improvements
of the beam maps and window functions. Details of the processing used to generate these
products are described in the remainder of this work.
In an accompanying paper Gold et al. (2010) utilize the maps in the five frequency
bands and some external data sets to estimate levels of Galactic emission in each map, and
describe the generation of a set of reduced foreground sky maps based on template cleaning,
and a map generated using an Internal Linear Combination map (ILC) of WMAP data, both
of which are used for analysis of the CMB anisotropy signal.
Larson et al. (2010) describe the measurement of the angular power spectrum of the
CMB obtained from the reduced foreground sky maps and the cosmological parameters
obtained by fitting the CMB power spectra to current cosmological models.
The cosmological implications of the data, including the use of external data sets, are
discussed by Komatsu et al. (2010), while Bennett et al. (2010) discuss a number of arguably
anomalous results detected in previous WMAP data releases.
Weiland et al. (2010) describe the characteristics of a group of point-like objects observed
by WMAP in the context of their use as microwave calibration sources for astronomical
observations.
The remainder of this paper is organized as follows: Section 2 presents updates on
the data processing procedures used to generate the seven-year sky maps and related data
products. Section 3 describes ongoing efforts to characterize the WMAP beam shapes,
while Section 4 presents the seven-year sky map data and power spectra, describes some
additional analyses on the low-ℓ polarization power spectra, and summarizes the scientific
results obtained from the latest WMAP data set.
– 4 –
2. Data Processing Updates
2.1. Operations
The sixth and seventh years of WMAP ’s operation span the interval from 00:00:00 UT,
9 August 2006 (day number 222) to 00:00:00 UT, 10 August 2008 (day number 222) and
includes nine short periods when observations were interrupted. These periods include eight
scheduled events: six station keeping maneuvers, and one maneuver to avoid flying through
the Earth’s shadow (11 November 2007), followed by a small orbital correction 19 days later.
The other interruption to observations occurred as a result of the failure of the primary
transmitter used to telemeter data to Earth. WMAP was subsequently reconfigured to use
its backup transmitter and normal operations resumed with no performance degradation.
On 1 August 2008 (day number 141) WMAP flew through the Moon’s shadow, causing a
≈ 4% decrease in the incident solar flux lasting ≈ 6h. WMAP remained in normal observing
mode throughout this event, but data was excluded from sky map processing due to minor
instrument thermal perturbations as described in § 2.3.
2.2. Calibration
The algorithm used to calibrate the time ordered data (TOD) is the same as was used
for the five-year processing (Hinshaw et al. 2009). Calibration occurs in two steps. First an
hourly absolute gain and baseline are determined. The absolute calibration is based on the
CMB monopole temperature (Mather et al. 1999) and the velocity dependent dipole resulting
from WMAP ’s orbit about the solar system barycenter. The calibration is performed
iteratively, since removing the fixed sky signals arising from the barycentric CMB signal and
foregrounds requires values of both the gain and baseline solutions. The only change made to
this step of the calibration procedure is that the initial value for the barycentric CMB dipole
signal has been updated to agree with the value determined from the five-year analysis.
The second step in the calibration procedure consists of fitting the hourly radiometer
gain values to a model which relates the gain to measured values of radiometer parameters.
The functional form of the gain model is the same as for the five-year data release, but the
fitting procedure now utilizes all seven years of observational data. The seven-year mean
changes in the calibration are presented in Table 1. Note that the rms of the fractional
calibrations change, ∆G/G, is only 0.05%. The absolute calibration accuracy is estimated
to be 0.2%, unchanged from the previously published value.
For K, Ka, and Q bands, there is a small but significant year-to-year variation in the
– 5 –
Table 1. Noise and Calibration Summary for the Template Cleaned and Uncleaned Maps
DA σ0(I) σ0(Q, U) σ0(Q, U) ∆G/G a
uncleaned template cleaned b
(mK) (mK) (mK) (%)
K1 1.437 1.456 NA -0.14
Ka1 1.470 1.490 2.192 -0.01
Q1 2.254 2.280 2.741 0.01
Q2 2.140 2.164 2.602 0.01
V1 3.319 3.348 3.567 -0.03
V2 2.955 2.979 3.174 -0.03
W1 5.906 5.940 6.195 -0.05
W2 6.572 6.612 6.896 -0.04
W3 6.941 6.983 7.283 -0.08
W4 6.778 6.840 7.134 -0.12
a∆G/G is the change in calibration of the current (seven-year) processing relative to the
five-year processing. A positive value means that features in the seven-year maps are larger
than the same features in the five-year map.
bThe σ0 value for the Stokes I template cleaned maps is the same as for the uncleaned
maps.
– 6 –
WMAP calibration that can be seen as variations of the Galactic plane brightness in yearly
sky maps. This has been measured by correlating each yearly map against the 7-year map
for pixels at |b| < 10. A slope and offset is fit to each correlation and the slope values are
adopted as yearly calibration variations. These are shown in Figure 1 and are consistent
with the 0.2% absolute calibration uncertainty.
2.3. Gaps in Observations
The WMAP instrument provides the highest quality data when the observatory is scan-
ning the sky in its nominal observing mode and the instrument thermal environment is
stable. Periodically, conditions arise that result in one or both of these conditions not be-
ing met. These situations arise from scheduled station-keeping maneuvers, and unscheduled
events, such as solar flares or on-board equipment malfunctions. Data taken during these
periods are excluded from the sky map processing to ensure the highest quality data prod-
ucts. Previously, potentially corrupted data segments were identified by manually inspecting
the event logs and instrument thermal trend plots. The current data release uses a more
objective automated procedure to identify the unusable data segments. The automated pro-
cedure was designed to approximate the manual procedure, but small differences will occur
in the sky coverage between the first five years of the current data release and the previous
five-year data release. Using the new procedure WMAP still achieves an overall observing
efficiency of ≈ 98.4%, down slightly from its previous value of ≈ 99%.
The automated procedure identifies suspect events based on the time-derivative of the
focal plane assembly (FPA) temperature. The measured FPA temperature is averaged over
one hour intervals, and the time derivative is formed by differencing successive values of these
averages. Suspect thermal events are delineated by the times at which the magnitude of this
derivative initially exceeds then finally falls below a threshold value. The threshold has been
chosen to be 5 times the rms deviation of the temperature derivative signal occurring during
normal observing periods, and corresponds to a value of ≈ 0.75 mK h−1. In situations where
the observatory was taken out of normal observing mode (e.g., during a scheduled station
keeping maneuver) the duration of the event is lengthened if needed to encompass the entire
time the observatory was not in normal observing mode. For each event data from 1.2
hours before the event began to 7.25 hours after the event ended is excluded from sky map
processing.
– 7 –
Fig. 1.— Measurements of the year-to-year calibration variation for K, Ka and Q bands
obtained by correlating the Galactic plane signal in the seven-year map to the signal in
single year sky maps. Note that the measured variations are consistent with the estimated
absolute calibration uncertainty of 0.2%. No significant variation is seen for the V and W
band maps.
– 8 –
2.4. Planet Masking
In the one, three and five-year data analyses, observations when the boresight of each
telescope beam fell within 1.5 of Mars, Jupiter, Saturn, Uranus or Neptune were excluded
from sky map processing, preventing contamination of the sky maps by emission from the
planets. Subsequent analysis has shown that even with these exclusion criteria microwave
emission from Jupiter could generate as much as a 75 µK (in K band) errant signal in a narrow
annulus surrounding the region of excluded observations. Although there is no indication
that this influenced the cosmological analysis, the radii used to exclude observations have
been increased for the seven-year analysis, chosen to limit planetary leakage to less than
1 µK. However, given the uncertainty in the determination of the beam profiles at these
levels, we adopt a conservative upper bound on possible planetary signal leakage into the
TOD used for sky map production of 5 µK. The new values are displayed in Table 2.
2.5. Expanded Diffuse Galactic Foreground Mask
The mask for diffuse Galactic emission has been revised by including areas near the
plane where the diffuse foreground cleaning algorithm appears to be less efficient than for
the sky at large. These areas are found by performing a pixel-by-pixel χ2 test comparing null
maps to cleaned Q-V and V-W maps with resolution degraded from r9 1 to r5. All pixels
with χ2 greater than 4 times that of the χ2 in the polar caps regions are cut. Small islands of
cut pixels are eliminated from the cut if they contain 4 or fewer contiguous pixels. The two
1WMAP uses the HEALPix (Gorski et al. 2005) pixelization and labels resolution as r4, r5, r9 and r10
corresponding to HEALPix Nside values of 16, 32, 512 and 1024 respectively.
Table 2. Data Exclusion Radii for the Planets
Frequency Band
Planet K Ka Q V W
Mars 2.0 1.5 1.5 1.5 1.5
Jupiter 3.0 2.5 2.5 2.2 2.0
Saturn 2.0 1.5 1.5 1.5 1.5
Uranus 2.0 1.5 1.5 1.5 1.5
Neptune 2.0 1.5 1.5 1.5 1.5
– 9 –
resulting masks based on Q-V and V-W analyses, respectively, are combined and promoted
back to r9. The edges of the cut are smoothed by convolving the mask with a Gaussian of
3 FWHM and cutting the result at a value of 0.5. As the final step, the smoothed cut is
combined with the five-year KQ85 or KQ75 cut (Gold et al. 2010). The resulting masks,
termed KQ85y7 and KQ75y7, admit 78.3% and 70.6% of the sky, respectively, as compared
to 81.7% and 71.6% for the five-year versions—a decrease of the admitted sky area by 3.4%
of the full sky for KQ85 and 1.0% for KQ75.
2.6. Map-Making with Asymmetric Masking
The most significant change in the current processing is the use of asymmetric masking
in the iterative reconstruction of the sky maps. WMAP ’s differential design means that the
time-ordered-data (TOD) represents differences between the intensities of pairs of points on
the sky observed by the two telescope beams. Reconstructing sky maps from differential data
requires solving a set of linear equations that describe the relation between the differential
TOD and the sky signal. Solution of this set of equations is performed iteratively, using sky
maps from earlier iterations to alternately remove estimated sky signals for each beam from
the TOD, leaving the value associated with the opposite beam which is then used to generate
the next iteration of the sky map. This procedure becomes problematic when one of the
beams is in a region of high intensity and the other is in a region of low intensity. Small errors
arising from pixelization, residual pointing errors, beam ellipticity or radiometer gain errors
limit the degree to which the signal associated with the beam in the high emission region
can be estimated. Such errors result in errant signals being introduced into sky map pixels
associated with the beam in the low intensity region. This potential problem is circumvented
by masking portions of the data when such errors are likely to be introduced.
In the WMAP first year results an asymmetric masking procedure was used (Hinshaw
et al. 2003). Asymmetric masking means that when one beam is in a high Galactic emission
region (as determined by a processing mask (Limon et al. 2010)) and the other beam is
in a low Galactic emission region, only the pixel in the high emission region is iteratively
updated. This allows for the reconstruction of full sky maps while avoiding the situation
that could produce an errant signal. The maps, t1, were obtained through a simple iterative
solution of the linear equation
t1 = Wd1 (1)
where d1 is the pre-whitened time ordered data and W = (MTM)−1 ·MT . The matrix M is
the mapping function, which has Np columns and Nt rows, corresponding to the number of
sky map pixels and time-ordered data points respectively. When the map processing includes
– 10 –
polarization degrees of freedom Np is four times the number of map pixels, corresponding
to the Stokes I, Q and U components and a spurious component, S, used to absorb effects
arising from bandpass differences between the two radiometers comprising each differencing
assembly (DA). See Jarosik et al. (2007) for a description of the implementation of the
spurious mode, S. Multiplying a sky map vector by M effectively generates the time ordered
data that would be obtained if WMAP observed a sky corresponding to the map vector.
The sky maps obtained by solving equation 1 are an unbiased representation of the true sky
signal, but do not treat the noise terms optimally.
The WMAP three-year (Jarosik et al. 2007) and five-year (Hinshaw et al. 2009) analyses
generated maximum likelihood map solutions using a conjugate gradient iterative technique.
The maximum likelihood estimate of the sky map, t, is
t = (MTN−1M)−1 · (MTN−1d) (2)
where d is the calibrated but unfiltered time ordered data, and N−1 is the inverse of the
radiometer noise covariance matrix. Solution of this equation involves a direct evaluation of
the last term of equation 2 once, resulting in a map t0 = MTN−1d, followed by the iterative
solution of the equation
t = (MTN−1M)−1 · t0. (3)
Solving this equation with the conjugate gradient method requires the use of symmetric
masking since this method can only be used when the matrix multiplying t0 (in this case
(MTN−1M)−1) is symmetric . Symmetric masking means that if either beam is in a region
of high Galactic emission, neither pixel is updated. Symmetric masking is implemented by
simply replacing all occurrences of the full sky mapping matrix, M, with a masked version
of the matrix in equations (2 and 3). Since symmetric masking produces sky maps with no
information in the high Galactic emission regions, in previous analyses two sets of sky maps
were generated, symmetrically masked maps, and full sky maps incorporating no masking.
Data from full sky maps were used to fill in regions of the symmetrically masked maps
which contained no data. This procedure was a significant advancement over the iterative
procedure used in the WMAP first year results, in that it produced maximum likelihood
maps and allowed for direct determination of the level of convergence of the solution. Its
major disadvantages are that two sets of maps have to be generated then pieced together,
and there is no simple method of calculating a pixel-pixel noise matrix which describes the
noise covariance between pixels obtained from the two different input maps. The pixel-pixel
noise covariance matrix delivered with the data release only applied to the pixels in the low-
Galactic emission regions, and only diagonal weights are used to describe the noise properties
in the regions of high Galactic emission.
– 11 –
Incorporating asymmetric masking into the maximum likelihood sky maps solution re-
quires solving the equation
t = (MT
amN−1M)−1 · (MT
amN−1d) (4)
where MTam is the mapping matrix incorporating asymmetric masking and M is the unmasked
mapping matrix. Again, this equation is solved by direct evaluation of the last term,
t0 = MT
amN−1d (5)
once, followed by an iterative solution of the equation
t = (MTamN−1M)−1 · t0. (6)
Note that the matrix MTamN−1M is not symmetric, and therefore this equation cannot be
solved using a simple conjugate gradient algorithm as was done previously. A bi-conjugate
gradient stabilized method (Barrett et al. 1994) was chosen to solve this equation since it of-
fered good convergence properties and was straightforward to implement, requiring relatively
minor changes to the existing procedures. This technique was verified by reconstructing input
maps to numerical precision from simulated noise-free TOD. Utilizing asymmetric masking
eliminates both of the aforementioned shortcomings of the simple conjugate gradient method
- each DA only requires a single map solution, and a pixel-pixel inverse noise covariance ma-
trix can be generated which describes the noise correlation between all the pixels in the
map.
2.7. Calculation of the pixel-pixel noise covariance matrix, Σ
The pixel-pixel noise covariance matrix is given by Σ = 〈tntTn 〉, where tn is the noise
component of a reconstructed sky map and the brackets denote an ensemble average. The
time-ordered data, d, used to generate the map may be written as the sum of a signal term
and a noise term
d = Mt + n (7)
where M is the full sky (unmasked) mapping matrix, t is a vector representing the input sky
signal and n is a vector corresponding to the radiometer noise, its covariance being described
as
N = 〈nnT〉. (8)
The noise component of the maximum likelihood asymmetrically masked map solution may
be written as
tn = (MT
amN−1M)−1 · (MT
amN−1n). (9)
– 12 –
The noise covariance matrix becomes
Σ = 〈(MTamN−1M)−1(MT
amN−1n) · [(MTamN−1M)−1(MT
amN−1n)]T 〉 (10)
= 〈(MTamN−1M)−1(MT
amN−1n) · (nT N−1Mam)(MTN−1Mam)−1〉 (11)
= (MTamN−1M)−1(MT
amN−1) · (〈nnT 〉N−1)Mam(MTN−1Mam)−1 (12)
= (MTamN−1M)−1 · (MT
amN−1Mam) · (MTN−1Mam)−1. (13)
In practice the terms MTamN−1M and MT
amN−1Mam are evaluated at r4 with the N−1 nor-
malized to unity at zero lag, and the inverse pixel-pixel noise covariance matrix is generated
by forming the product
Σ−1 = (MTN−1Mam) · (MTamN−1Mam)−1 · (MT
amN−1M). (14)
Regions of the Σ−1 matrix corresponding to combinations of Stokes I, Q and U are then
converted to noise units by dividing by the appropriate combinations of σ0(I) and σ0(Q, U)
given in Table 1.
The term MTamN−1Mam is numerically inverted and is used as the source for the off-
diagonal terms of the preconditioner for the bi-conjugate gradient stabilized algorithm.
2.8. Nobs Fields of the Maps
The asymmetric masking used to generate sky maps requires a new procedure for cal-
culating the effective number of observations, Nobs, for each map pixel. In previous data
releases the effective number of observations was calculated by accumulating the number of
time ordered data points falling within each pixel, weighted by the appropriate transmission
imbalance coefficients, xim, and polarization projection factors (sin 2γ, cos 2γ). These val-
ues corresponded to the diagonal elements of the inverse pixel-pixel noise covariance matrix
Σ−1 = MT N−1M (Jarosik et al. (2007) equation (26)), evaluated assuming white radiometer
noise, i.e. N−1 = I. The simple procedure for evaluating these terms is not applicable to the
form of Σ−1 for the asymmetrically masked maps, equation (14). The Nobs fields of the r9
and r10 sky maps were generated by evaluating the diagonal elements of equation (14) with
N−1 = I using sparse matrix techniques. The Nobs fields of the r4 sky maps are described
in § 2.10.
2.9. Projecting Transmission Imbalance Modes from the Σ−1 Matrices
As described in Jarosik et al. (2007), errors in the determination of the transmission im-
balance parameters, xim, are a potential source of systematic artifacts in the reconstructed
– 13 –
sky maps. These time independent parameters, which specify the difference between the
transmission between the A-side and B-side optical systems for each radiometer, are mea-
sured from the flight data, and are presented in Table 3. To prevent biasing the cosmological
analyses, sky map modes that can be excited by these measurement errors are identified and
projected from the Σ−1 matrices.
The procedure follows the same method as used in the three and five years analyses
and consists of generating a one-year span of simulated time ordered data using the nominal
values of the loss imbalance parameters. Sky maps are processed from this archive using
the input values of xim and altered values to simulate errors in the measured values of the
coefficients. The differences between the resultant maps are used to identify map modes
resulting from processing the data with altered xim values. Two modifications have been
made to this procedure relative to the previous analyses: (1) The simulated sky maps are
formed at r4 by direct evaluation of equations (5, 6) at r4. This change was adopted to
eliminate contamination of the transmission imbalance templates by the poorly measured
sky map modes associated with monopoles in the Stokes I and spurious mode S sky maps.
It was found that varying the xim factors produced large changes in the amplitudes of these
poorly measured modes in addition to exciting the mode associated with the loss imbalance.
Through utilization of a singular value decomposition it was possible to null the poorly
measured modes associated with the aforementioned monopoles while preserving the modes
associated with the transmission imbalance while evaluating equation (6). (2) Transmission
imbalance modes were evaluated both for the case when the xim for both radiometers com-
prising each DA were increased 20% above their measured values, and for the case when the
xim value for radiometer 1 was increased by 10% and that of radiometer 2 decreased by 10%.
(Previously the only combination used was that in which both xim values were increased.)
For six of the DAs very similar sky map modes were generated by both sets of xim, while
for the V2, W1, W2, and W4 somewhat different modes were generated for the two different
sets. As a result, only one mode was projected from the K1, Ka1, Q1, Q2, and W3 matrices
while two modes were projected out of the the V2, W1, W2 and W4 matrices. In each case
the modes were removed from Σ−1following the method described in Jarosik et al. (2007).
2.10. Low Resolution Single Year Map Generation
The low resolution sky maps were generated by performing an inverse noise weighted
degradation of the high resolution maps that takes into account the intra-pixel noise corre-
lations of the high resolution input maps . The weight matrix for the polarization (Stokes
– 14 –
Q and U) and spurious mode (S) of each high resolution map pixel, p9, is given by
Nobs(p9) =
NQQ NQU NQS
NQU NUU NSU
NQS NSU NSS
, (15)
where each element NXY is an element of Σ−1 (equation 14) evaluated as described in § 2.8.
For each DA year combination the Q, U, S map sets were generated as
Qp4
Up4
Sp4
=(
Ntotobs(p4)
)−1∑
p9∈p4
Nobs(p9)
Qp9
Up9
Sp9
, (16)
Ntotobs(p4) =
∑
p9∈p4
Nobs(p9). (17)
The Nobs fields of the low resolution sky maps contain the diagonal elements of the cor-
responding portions of the Σ−1 matrices as described in § 2.7 before the scaling by σ0 is
applied.
2.11. Low Resolution Multi-year Map Generation
The single year single DA maps were combined to form a seven-year map for each DA
and seven-year maps for each frequency band. The individual low resolution maps were
inverse noise weighted using the Σ matrices (§ 2.9), scaled to reflect the noise level of each
DA. The weighted polarization maps were formed as
(
Q
U
)
= Σtot∑
yr,DA
Σ−1
yr,DA
(
Q
U
)
yr,DA
, (18)
Σtot =
(
∑
yr,DA
Σ−1
yr,DA
)−
, (19)
where ()− represents a pseudo-inverse of the sum in parenthesis. This summation is per-
formed over the year/DA combinations to be included in the final map. For K and Ka-bands
the pseudo-inverse is calculated by performing a singular value decomposition of the sum of
the QU × QU Σ−1 matrices and inverting all its eigenvalues except for the smallest, which
is set to zero. This procedure de-weights the one nearly singular mode associated with the
monopoles in the I and S maps. No modes are removed from the Q, V and W-band maps.
– 15 –
3. Beam Maps and Window Functions
The seven-year WMAP beams and window functions result from a refinement of the
data reduction methods used for the five-year beams, with no major changes in the processing
steps.
Briefly, in the five-year analysis, the beam maps were accumulated from TOD samples
with Jupiter in either the A or the B side beam. Sky background was subtracted using the
seven-year full-sky maps, with the band dependent Jupiter exclusion radii (§2.4). Because
of the motion of Jupiter, and the numerous Jupiter observing seasons2, the sky coverage of
the maps was 100% in spite of this masking.
As far as possible, the beam transfer functions bℓ in Fourier space were computed directly
from Jupiter data. However, at low signal levels, some information was incorporated from
beam models. The A and B side Jupiter data were fitted separately by a physical optics
model comprising feed horns with fixed profiles, together with primary and secondary mirrors
described by fit parameters. The geometrical configuration of these components was fixed.
The fitted parameters described small distortions of the mirror surface shapes.
The beam models were combined with the Jupiter data at low signal levels by a hy-
bridization algorithm. This process was optimized to yield the minimum uncertainty in
beam solid angle, given the instrumental noise, under the conservative assumption that the
systematic error in the beam model was 100%. Details, as well as a summary of beam
processing in earlier data releases, were given by Hill et al. (2009).
In addition to the use of three more seasons of Jupiter data, the following refinements,
described below in more detail, have been made for the seven-year beam processing:
• Subtraction of background flux from Jupiter observations is improved by expanding
the exclusion radius for sky maps (§ 2.4)
• The beam modeling procedure was modified by adding additional terms to the descrip-
tion of the secondary mirror figure to approximate a hypothetical tip/tilt.
• The rate of convergence of the physical optics models is improved by correcting an
orthogonalization error in the conjugate gradient code.
• The physical optics fit is driven harder in two senses:
2A “season” is a ≈ 45 day period when a planet falls within the scan pattern of WMAP . Each planet
typically has two seasons per year.
– 16 –
1. A smaller change in χ2 between conjugate gradient descent steps is required to
declare convergence.
2. The model is fitted over a wider field of view in the observed beam map.
The change in beam profiles and transfer functions between the five- and seven-year
analyses is conveniently summarized by a comparison of solid angles for the 10 DAs, given
in Table 4. The solid angle increases of 0.8% in K1 and 0.4% in Ka1 are the result of the
improved background estimates for the Jupiter observations. The solid angle changes for
the Q1–W4 DAs have multiple causes. First, the instrumental noise in the Jupiter samples
is different in detail. Second, the beam models have been refitted and differ slightly from
the five-year versions. Third, the increased S/N of the Jupiter data in the beam wings
means that for some DAs, the hybrid threshold is optimized to a lower gain with 7 years
of data. This value is the limit below which observed intensity values in the Jupiter TOD
are replaced with computed values from the beam model. For the 5 year data analysis, the
hybrid thresholds were 3, 4, 6, 8, and 11 dBi, respectively, for bands K, Ka, Q, V, and W
(Hill et al. 2009). For the seven-year data analysis, the V and W thresholds are 7 and 10
dBi, respectively, while those for K, Ka, and Q are unchanged.
As shown in the table, the aggregate solid angle changes for V and W, which are the
bands used in the high-ℓ TT power spectrum, are ∼ 0.1% or less. Table 4 also gives the cur-
rent values of the forward gain Gm and of the factor Γff for converting antenna temperature
to flux in Jy. New hybrid beam profiles and beam transfer functions bℓ are available online
from the Legacy Archive for Microwave Background Data Analysis (LAMBDA).
3.1. Seven-year Beam Model Fitting
3.1.1. Secondary Tip/Tilt
One change in the seven-year beam computations is to introduce approximate tip/tilt
terms for the secondary mirror into the physical optics model. This change is motivated
by the stable ∼ 0.1 offset in the collective B-side boresight pointings, as compared to pre-
flight expectations (Hill et al. 2009). In previous beam fits, this offset was absorbed by two
free-floating nuisance parameters.
The introduction of secondary tip/tilt is a heuristic attempt to improve the fidelity of
the beam model. Because a displacement in beam pointings could result from various small
mechanical displacements in the instrument, or from a combination of them, the beams
contain too little information to support a mechanical analysis. We emphasize that the
– 17 –
boresight pointings on both the A and the B sides are equally stable since launch, to < 10′′
(Jarosik et al. 2007), an estimate that is confirmed by the full seven-year Jupiter data.
For convenience, we approximate tip/tilt as a purely planar surface distortion of the
secondary mirror, in addition to the Bessel modes used previously. The pivot line is allowed
to vary by including a term for scalar displacement together with tip/tilt slopes in two
orthogonal directions.
This parameter subspace has been explored using Monte Carlo beam simulations, which
reveal a strong degeneracy between the scalar offset and a tilt in the spacecraft Y Z plane,
because both displacements move the illumination spot on the primary mirror in the up or
down direction. Therefore, we have investigated a family of solutions by adopting an Ansatz
for the mechanical constraints, such that (1) one edge of the secondary is allowed to pivot
while pinned at the designed position, (2) the opposite edge undergoes a small displacement,
and (3) the secondary mirror as a whole is rigid.
These constraints result in two discrete solutions for each secondary mirror; for each side
of the instrument, the solution requiring the smaller mechanical displacement is chosen. The
resulting edge displacement is −4.4 mm on the B side, which corresponds to a hypothetical
motion of one edge of the secondary mirror away from the secondary backing structure and
toward the primary. Similarly, this method results in a hypothetical displacement of one edge
of the A-side secondary by 0.2 mm away from the primary. However, as already explained,
this mechanical interpretation is uncertain. For subsequent stages of beam modeling, the
distortion coefficients mimicking tip/tilt and displacement of the secondary mirrors are held
fixed.
3.1.2. Fitting Process
The overall fitting of beams progresses much as it did for the five-year data. On each of
the A and B sides, the primary mirror is modeled with Fourier modes added to the nominal
mirror figure. Similarly, the secondary mirror is modeled with Bessel modes. A complete fit
is done using modes with spatial frequency on the mirror surface up to some maximum value.
When convergence is achieved, the current best-fit parameters become the starting point for
a new fit with a higher spatial frequency limit. The primary mirror spatial frequencies are
indexed using an integer k, such that the wavelength of the mirror surface distortion is
280 cm/k. The secondary modes are specified by the Bessel mode indexes n and k. Details
are given in Hill et al. (2009), §2.2.2. The maximum spatial frequencies fitted are defined by
kmax = 24 on the primary and nmax = 2 on the secondary.
– 18 –
Because the Fourier modes are non-orthogonal over the circular domain of the primary
mirror distortions, the fitting is actually done in an orthogonalized space (Hill et al. 2009).
A bug in the orthogonalization code was corrected before the seven-year analysis began;
this bug only affected speed of convergence, not the definition of χ2, so the five-year results
remain valid. However, this change made it practical to refit the beam models ab initio from
the nominal mirror figures, i.e., with zero for all Fourier and Bessel coefficients, rather than
beginning from the five-year solution.
The χ2 changes from the five-year to the seven-year beam models are shown in Table
5. The χ2 values shown for the five-year models are recomputed using seven-year data. It is
clear from Table 5 that the overall improvement in the model for the 10 DAs collectively is
driven primarily by the W band on the A side and the V and W bands on the B side.
Figure 2 shows model beam profiles from an example DA, W1, from both the five-year
and the seven-year analyses. The A-side model has a similar profile in the five-year and
seven-year fits, with small trade-offs in fit quality at various radii. However, the B-side
model shows a new feature in the seven-year fitting, namely, an elevated response at large
radii and very low signal levels in the V and W bands. This feature is seen at radii larger
than 2 in the right-hand panel of Figure 2, where the seven-year profile (red) is very slightly
higher than the five-year profile (blue). The five-year and seven-year profiles for the example
DA differ by a factor of ∼ 2 − 10 in this region, although both are & 50 dB down from the
beam peak.
Several attempts have been made using jumps in the fitting parameters (as in simulated
annealing) to find a region of parameter space in the A-side fit that would lead to the
existence of elevated tails similar to those on the B side. So far, such attempts have failed.
The best A-side fit lacks the elevated-tail feature, while the best B-side fit has it.
Just as in the five-year analysis, none of the beam models meets a formal criterion for
a good fit, since the number of degrees of freedom is of order 105, or 104 per DA, while the
overall χ2ν is ∼ 1.05. However, the model beams are used only in a restricted role, more
than ∼ 45 dB below the beam peak, so that the primary contribution to the beams and bℓ is
directly from Jupiter data. As a result, the effect of any particular feature of the models is
either omitted or diluted in the final result. However, the B-side model tail mentioned above
increases the beam uncertainty slightly at large radii as compared to the five-year analysis.
The process for incorporating information from the models into the Jupiter-based beams
is explained by Hill et al. (2009). This process is termed hybridization. Briefly, the beam pro-
files are integrated from Jupiter TOD. The two-dimensional coordinates of each TOD point
within the beam model are determined. If the model gain is below a certain threshold, then
– 19 –
the model gain is substituted for the Jupiter measurement. The thresholds are optimized for
each DA by minimizing the uncertainty in the solid angle, under the assumption that the
beam model is subject to a scaling uncertainty of 100%. Final hybrid profiles combining the
A and B sides are shown for the W1 DA, for both the five-year and the seven-year analysis,
in Figure 3.
The beam transfer function, bℓ, is integrated directly from the hybrid beam profiles, and
the error envelope for bℓ is computed using a Monte Carlo method. These procedures are the
same as in the five-year analysis (Hill et al. 2009). A comparison of five-year and seven-year
bℓ and the corresponding uncertainties is shown in Figure 4.
3.2. Flux Conversion Factors and Beam Solid Angles for Point Sources
The conversion factor from peak antenna temperature to flux density for a point source
is given by (Page et al. 2003a)
Γ = c2/2kBΩeff(νeff )2 (20)
where Ωeff is the effective beam solid angle of the A and B sides combined and νeff is the
effective band center frequency, both of which depend on the source spectrum. New values of
these quantities have been calculated that supersede previous results given for point sources.
For a point source with antenna temperature spectrum TA ∝ νβ, the effective frequency
is determined from
νβeff =
∫
f(ν)Gm(ν)νβdν/
∫
f(ν)Gm(ν)dν (21)
where f(ν) is the passband response, and Gm(ν) is the forward gain. This is consistent with
the definition used by Jarosik et al. (2003) for a beam-filling source, except in that case
the forward gain is not included. Values of νeff (β) are calculated using pre-flight bandpass
measurements and pre-flight GEMAC3 measurements of forward gain. A correction for
scattering is applied to the GEMAC measurements,
Gm(ν) = GGEMACm (ν)e(−(4πσ/λ)2) (22)
where σ is an effective rms primary mirror deformation whose value is set separately for each
DA based on the degree of scattering estimated from the pass 4 physical optics modeling.
3Goddard Electromagnetic Anechoic Chamber
– 20 –
Fig. 2.— Physical optics beam models for the W1 DA on the A side (left) and B side
(right) of the WMAP instrument. The ordinate is scaled by a sinh−1 function that provides
a smooth transition between linear and logarithmic regimes. Blue: five-year models; red :
seven-year models. Points : seven-year Jupiter beam data averaged in radial bins of ∆r = 0.′5.
Dashed lines : Radii at which hybrid beam profiles consist of 90%, 50%, and 10% Jupiter
data, respectively, from smaller to larger radii. Model differences inside r ∼ 1.7 are mostly
suppressed in the hybrid beam profiles, whereas model differences outside r ∼ 2.6 are mostly
retained.
– 21 –
Fig. 3.— W1 hybrid beam profiles from five-year (blue) and seven-year (red) analysis,
combined for the A and B sides. The ordinate is scaled by a sinh−1 function that provides
a smooth transition between linear and logarithmic regimes. Dashed lines : Radii at which
hybrid beam profiles consist of 90%, 50%, and 10% Jupiter data, respectively, from smaller
to larger radii. The noise shows that use of Jupiter data extends effectively to larger radii
in the seven-year analysis.
– 22 –
Fig. 4.— Comparison of beam transfer functions and uncertainties between five-year and seven-year
analyses. Black : Relative change in beam transfer functions from five to seven years in the sense (7 yr −
5 yr)/(5 yr). Green: five-year 1σ error envelope. Red : seven-year 1σ error envelope. The seven-year bℓ
are largely within 1σ of the five-year bℓ, while the change in the error envelope itself is small. In W band,
modeling differences between the A and B sides introduce an increase in the uncertainty plateau for multipoles
ℓ . 1000, whereas the small angle (high ℓ) uncertainty is decreased for all bands.
– 23 –
The effective solid angle is calculated as
Ωeff = Ω(Jupiter)ΩGEMAC(β)/ΩGEMAC(Jupiter) (23)
where Ω(Jupiter) is the pass 4 beam solid angle determined from Jupiter observations
and Ω(GEMAC) is the beam solid angle calculated for a given source spectrum using the
scattering-corrected GEMAC measurements. From eq (25) of Page et al. (2003b),
Ω(GEMAC) = 4π
∫
f(ν)νβdν/
∫
f(ν)Gm(ν)νβdν (24)
Values of νeff , Ωeff , and Γ for a point source with β = −2.1, typical for the sources in
the WMAP point source catalog, are given in Table 2.
4. Sky Map Data and Analysis
The seven-year sky maps are consistent with the 5 year maps apart from small effects
related to the new processing methods. Figure 5 displays the seven-year band average Stokes
I maps and the differences between these maps and the published five-year maps. The
difference maps have been adjusted to compensate for the slightly different gain calibrations
and dipole signals used in the different analysis. The small Galactic plane features in the
K, Ka and Q band difference maps arise from the slightly different calibrations and small
changes in the effective beam shapes. Pixels in the Galactic plane region are observed over a
slightly smaller range of azimuthal beam orientation in the current data processing relative to
previous analyses, resulting in slightly less azimuthal averaging and hence a slightly altered
effective beam shape. The calibration gain changes relative to the five-year data release are
small, all below 0.15% as indicated in Table 1.
4.1. The Temperature and Polarization Power Spectra
4.1.1. The Temperature Dipole and Quadrupole
The five-year CMB dipole value was obtained using a Gibbs sampling method (Hinshaw
et al. 2009) to estimate the dipole signal in both the five-year Internal Linear Combination
map (ILC) and foreground reduced maps. The uncertainties on the measured parameters
were set to encompass both results as an estimate of the effect of residual foreground sig-
nals. The dipole measured from the seven-year data shows no significant changes from that
– 24 –
Fig. 5.— Plots of the Stokes I maps in Galactic coordinates. The left column displays the seven-year
average maps, all of which have a common dipole signal removed. The right column displays the difference
between the seven-year average maps and the previously published five-year average maps, adjusted to take
into account the slightly different dipoles subtracted in the seven-year and five-year analyses and the slightly
differing calibrations. All maps have been smoothed with a 1 FWHM Gaussian kernel. The small Galactic
plane signal in the difference maps arises from the difference in calibration (0.1%) and beam symmetrization
between the five-year and seven-year processing. Note that the temperature scale has been expanded by a
factor of 20 for the difference maps.
– 25 –
obtained from the five-year data, so the best fit dipole parameters remain unchanged from
the five-year values, and are presented in Table 6.
The maximum likelihood value of magnitude of the CMB quadrupole is l(l +1)Cl/2π =
197+2972−155 µK2 (95% CL) (Larson et al. 2010) based on analysis of the ILC map using the
KQ85 mask. This value is essentially unchanged from the five-year results and lies below
the most likely value predicted by the best fit ΛCDM model. This value, however, is not
particularly unlikely given the distribution of values predicted by the model, as described in
Bennett et al. (2010).
Figure 6 displays the seven-year band average maps for Stokes Q and U components
for all 5 WMAP frequency bands. Polarized Galactic emission is evident in all frequency
bands. The smooth large angular scale features visible in the W-band maps, and to a lesser
degree in the V-band maps, are the result of a pair of modes that are poorly constrained
by the map-making procedure. While these dominate the appearance of the map, they are
properly de-weighted when these maps are analyzed using their corresponding Σ−1 matrices,
so useful polarization power spectra may be obtained from these maps. The relatively large
amplitudes of these modes limits the utility of using difference plots between the five-year
and seven-year map sets to test for consistency.
4.1.2. Low-ℓ W-band Polarization Spectra
Previous analyses (Hinshaw et al. 2009; Page et al. 2007) exhibited unexplained artifacts
in the low-ℓ W-band polarization power spectra demonstrating an incomplete understanding
of the signal and/or noise proprieties of these spectra. Specifically the value of CEEℓ for ℓ = 7
measured in W-band was found to be significantly higher than could be accommodated by
the best fit power spectra, given the measurement uncertainty. This result, and several other
anomalies, caused these data to be excluded from cosmological analyses. Significant effort
has been expended trying to understand these spectra with the goal of eventually allowing
their use in cosmological analyses.
A set of null spectra were formed based on the latest uncleaned W-band polarization sky
maps to test for year-to-year and DA-to-DA consistency. Polarization cross power spectra
were calculated for pairs of maps using the Master algorithm (Hivon et al. 2002) utilizing
the full Σ−1 covariance matrix to weight the input maps. The polarization analysis mask
was applied by marginalizing the Σ−1 over pixels excluded by the mask to minimize fore-
ground contamination. Appropriately weighted null signal combinations of these spectra
were formed to determine if any individual years or DAs possessed peculiar characteristics.
– 26 –
Fig. 6.— Plots of the Seven-year average Stokes Q and U maps in Galactic coordinates. All maps have
been smoothed with a 2 FWHM Gaussian kernel.
– 27 –
The uncertainties on the power spectra were evaluated using the Fisher matrix technique
(Page et al. 2007) and measured map noise levels. Since the input sky maps contain both
signal (mostly of Galactic origin) and noise, an additional term was added to the Fisher
matrix noise estimate to account for the signal × noise cross term. The signal component of
this term was estimated using the seven-year average power spectra values for the combined
W1, W2, and W3 DAs. This term was only added for multipole/polarization combinations
(EE or BB) for which the estimated signal was greater than 0.
Table 7 shows the result of this analysis. The reduced χ2 combinations in the top panel
are evaluated for data combinations of all 4 W-band DAs that compare individual years to
the average of the remaining years. Polarization combinations EE and BB were evaluated
for multipoles ranging from 2-32. Additional combinations were formed to compare the first
three years of data to the latter four, and to compare data taken in odd and even numbered
years. All combinations resulted in reasonable χ2 and probability to exceed values.
The lower half of the table contains the results of a similar analysis, but in this case
combinations were formed to isolate individual DAs. The W4 is singled out with a reduced
χ2 value of 1.163 which has only a probability to exceed of only 0.05%. This DA has an
unusually large 1/f knee frequency, which makes it particularly susceptible to systematic
artifacts. For this reason it is excluded from the analysis below, but continues to be studied
in case it might contain clues as to the nature of the low-ℓ polarization anomalies seen in
the other W-band DAs.
Figure 7 displays polarization power spectra for the EE, BB and EB modes for the
first three, five, and seven years of template cleaned individual year maps from the W1, W2
and W3 DAs. These spectra were also obtained using the Master algorithm utilizing the
mask-marginalized Σ−1 covariance matrix sky map weighting. Only cross power spectra are
included, so the measurement noise does not bias the measured values. The uncertainties
are based on the Fisher matrix technique, and have not had the signal × noise term added,
since any signal remaining in the cleaned maps is expected to be small. (Master algorithm
based low-ℓ power spectra are not used in any cosmological analyses.)
There is generally good agreement between the values for the three different time ranges.
Note that the high value seen in the three and five-year analysis for CEE7 has fallen signif-
icantly with the additional two years of data. In fact, had the data been take in reverse
order (starting with year seven) the CEE7 value would not have been identified as particularly
anomalous. However, the CEE12 value has risen in the seven-year combination. To see if these
signals might be artifacts of the foreground cleaning, a similar analysis was performed on
the uncleaned sky maps and is displayed in Figure 8. By comparing the two figures it can be
seen that the foreground cleaning mainly affects the multipoles ℓ ≤ 5 leaving the ℓ = 7, 12
– 28 –
Fig. 7.— Low ℓ Master polarization cross power spectra from the first three, five, and seven
years of data from the W1, W2 and W3 foregrounds reduced polarization sky maps (Gold
et al. 2010) for the first three, five and seven years of observations. These three time ranges
contain 36, 105 and 210 individual cross spectra respectively. The ℓ values for the different
time ranges have been offset for clarity. The top three panels contain the Master power
spectra and error bars based on a Fisher matrix analysis. The bottom panel is the ratio
of the measured variance between the individual power spectra estimates ( DA year × DA
year ) and the variance predicted by the Fisher matrix calculation using the full Σ−1 inverse
noise covariance matrix. Note the good agreement for ℓ ≥ 8.
– 29 –
multipoles relatively unchanged.
The bottom panel of these figures plot the ratio between the variance measured among
the individual cross spectra for each multipole/polarization combination, to that predicted
by the Fisher matrix technique. For ℓ ≥ 8 there is good agreement between the measured
variance and analytical estimate, but the measured variance slowly grows larger with de-
creasing ℓ for ℓ ≤ 7 for both the cleaned and uncleaned sky maps. The Fisher matrix noise
estimate utilizes the Σ−1 matrix that describes the noise correlation between the pixels of
the Stokes Q and U maps. This matrix incorporates information regarding the sky scanning
pattern, through the mapping matrices M and Mam, and correlations in the radiometer
noise, through the use of inverse radiometer noise covariance matrix N−1. This method does
not, however, include the effect of any noise correlations that might be introduced through
the baseline fitting and calibration procedure. A program was undertaken to determine if
the baseline fitting and calibration procedure could be affecting the low-ℓ polarization results
and is described in the following section.
4.1.3. Calibration Induced Spectral Uncertainties
Uncertainties in the polarization power spectra arising from calibration of the TOD were
evaluated using the Fisher information matrix method . New sets of noise matrices were cal-
culated including degrees of freedom associated with the gain and baseline parameters used
in the calibration procedure. The values of these two parameters were approximated as
piecewise functions, each assumed constant over one hour intervals. The calibration proce-
dure was modeled as a χ2 minimization of the residuals between the TOD and a predicted
TOD calculated by applying the gain and baseline values for each time interval to a model
signal consisting of CMB anisotropy and a time dependent dipole due to WMAP ’s mo-
tion about the solar system barycenter. Fisher information matrices were calculated for
this model and inverted to form noise matrices. Uncertainties in the recovered low-ℓ power
spectra were calculated using this method by marginalizing over the values of the gain and
baseline parameters. A similar calculation was performed with the gain and baseline terms
omitted. The effects of the calibration on the uncertainties in the recovered power spectra
were measured by comparing the results of these two calculations.
As expected, the difference in the predicted uncertainties was small for all but the very
lowest multipoles. This occurs because the signal from the low-ℓ multipoles enters the TOD
on the longest time scales, and therefore is most affected by a the calibration procedure
which fits the low frequency dipole signal used for calibration. For E-mode polarization the
maximum increase in uncertainty in alm is about 3.5% occurring for ℓ = 3 and is less than
– 30 –
Fig. 8.— Low ℓ Master polarization cross power spectra from the first three, five, and seven
years of data from the W1, W2 and W3 uncleaned sky maps. The ℓ values have been
offset slightly for the different time ranges for clarity. The bottom panel is the ratio of the
measured variance between the individual power spectra estimates ( DA year × DA year )
and the variance predicted from a Fisher matrix calculation using the full Σ−1 inverse noise
covariance matrix. Note the good agreement for ℓ ≥ 8.
– 31 –
1% for other multipoles. For the B-mode alm uncertainties for ℓ = 2, 3 increase by 40% and
90% respectively, while the effect on other multipoles is less that 1%. The ℓ = 3 B-mode
polarization was already known to be very poorly measured by WMAP , since its symmetry,
combined with WMAP ’s geometry and scan pattern, generate extremely long period signals
(periods exceeding 10 minutes) in the TOD. The fact that the calculation described above
correctly identified this mode supports the validity of the methodology used, but the overall
results do no fully explain the excess variance observed in all the W-band ℓ ≤ 7 polarization
multipoles. The low-ℓ W-band polarization data therefore continue to be excluded from
cosmological analysis. However, there is no evidence suggesting any compromise of the high-
ℓ W-band polarization data, so it it now included in evaluation of the high-ℓ TE power
spectrum.
4.2. Science Highlights
The WMAP data remains one of the cornerstone data sets used for testing the cosmo-
logical models and the precision measurement of their parameters. Figure 9 displays the
binned TT and TE angular power spectra measured from the seven-year WMAP data (Lar-
son et al. 2010), along with the predicted spectrum for the best fit minimal six-parameter
flat ΛCDM model. The overall agreement is excellent, supporting the validity of this model.
Table 8 tabulates the parameter values for this model using WMAP data alone, and in com-
bination with other data sets. Details of the methodology used to determine these values
are described in Larson et al. (2010) and Komatsu et al. (2010).
The seven-year WMAP results significantly reduce the uncertainties for numerous cos-
mological parameters relative to the five-year results. The uncertainties in the densities of
baryonic and dark matter are reduced by 10% and 13% respectively. When tensor modes are
included, the upper bound to their amplitude, determined using WMAP data alone, is nearly
20% lower. By combining WMAP data with the latest distance measurements from Baryon
Acoustic Oscillations (BAO) in the distribution of galaxies (Percival et al. 2009) and Hub-
ble constant measurements (Riess et al. 2009), the spectral index of the power spectrum of
primordial curvature perturbations is ns = 0.963±0.012, excluding the Harrison-Zel’dovich-
Peebles spectrum by more than 3σ.
The reduced noise obtained by using the seven-year data set yields a better measure-
ment of the third acoustic peak in the temperature power spectrum. This measurement,
when combined with external data sets, leads to better determinations of the total mass of
neutrinos,∑
mν , and the effective number of neutrino species, Neff , as presented in Table 8.
Additionally, when augmented by the small scale CMB anisotropy measurements by ACBAR
– 32 –
(Reichardt et al. 2009) and QUaD (Brown et al. 2009), this result yields a greater that 3σ
detection of the primordial Helium abundance, YHe = 0.326± 0.075, using CMB data alone.
Komatsu et al. (2010) also demonstrate that, with the larger data set, the expected radial
and azimuthal polarization patterns around hot and cold peaks in the CMB can now be
observed directly in pixel-space by stacking sky map data. In addition, they now detect the
Sunyaev-Zel’dovich effect at ≈ 8σ at the location of known galaxy clusters, as determined
by ROSAT.
Finally, Weiland et al. (2010) have measured the brightness temperature of Jupiter,
Saturn, Mars, Uranus and Neptune, and five fixed calibrations objects, in all five frequency
bands, allowing their use as mm-wave celestial calibration sources traceable to WMAP ’s
precise calibration.
5. Summary
The WMAP observatory has successfully completed seven years of observations with
no significant performance degradation. A full set of sky maps for the seven year data span
has been generated and are available for analyses by the astrophysical community. These
maps were generated with an updated masking procedure that simplifies the map-making
procedure and allows creation of a single full-sky noise correlation matrix describing the
noise correlation over the entire sky for the reduced resolution sky maps. The understanding
of the beam profiles and resulting window functions has been improved with the additional
bean profile information obtained by more observations of Jupiter. The planetary observa-
tions, combined with the precise absolute calibration of WMAP , were used to measure the
brightness temperature of Mars, Jupiter, Saturn, Uranus and Neptune for use as calibration
sources.
Finally, the additional data and a better understanding of the instrument’s character-
istics has resulted in tighter constraints on the value of parameters of cosmological models.
WMAP is funded by the Science Mission Directorate Office at NASA Headquarters. We
acknowledge use of the HEALPix (Gorski et al. 2005) software package. The numerous data
products described in this document are available from the Legacy Archive for Microwave
Background Data Analysis (LAMBDA) http://lambda.gsfc.nasa.gov.
– 33 –
Fig. 9.— The temperature (TT) and temperature-polarization(TE) power spectra for the
seven-year WMAP data set. The solid lines show the predicted spectrum for the best-fit flat
ΛCDM model. The error bars on the data points represent measurement errors while the
shaded region indicates the uncertainty in the model spectrum arising from cosmic variance.
The model parameters are: Ωbh2 = 0.02260 ± 0.00053, Ωch
2 = 0.1123 ± 0.0035, ΩΛ =
0.728+0.015−0.016, ns = 0.963 ± 0.012, τ = 0.087 ± 0.014 and σ8 = 0.809 ± 0.024.
– 34 –
Table 3. Seven-year Transmission Imbalance Coefficients
Radiometer xim Uncertainty Radiometer xim Uncertainty
K11 -0.00063 0.00022 K12 0.00539 0.00010
Ka11 0.00344 0.00017 Ka12 0.00153 0.00011
Q11 0.00009 0.00051 Q12 0.00393 0.00025
Q21 0.00731 0.00058 Q22 0.01090 0.00116
V11 0.00060 0.00025 V12 0.00253 0.00068
V21 0.00378 0.00033 V22 0.00331 0.00106
W11 0.00924 0.00207 W12 0.00145 0.00046
W21 0.00857 0.00227 W22 0.01167 0.00154
W31 -0.00073 0.00062 W32 0.00465 0.00054
W41 0.02314 0.00461 W42 0.02026 0.00246
Note. — Transmission imbalance coefficients, xim, and their uncertainties determined from
the seven-year observational data.
– 35 –
Table 4. WMAP Seven-year Main Beam Parameters
ΩS7yr
a ∆(ΩS7yr)/ΩSb ΩS
7yr
ΩS5yr
− 1c Gmd νff
effΩff
effΓff
e
DA (sr) (%) (%) (dBi) (GHz ) (sr) (µK Jy−1)
For 10 Maps
K1 2.466 × 10−4 0.6 0.8 47.07 22.72 2.519 × 10−4 250.3
Ka1 1.442 × 10−4 0.5 0.4 49.40 32.98 1.464 × 10−4 204.5
Q1 8.832 × 10−5 0.6 -0.1 51.53 40.77 8.952 × 10−5 218.8
Q2 9.123 × 10−5 0.5 -0.2 51.39 40.56 9.244 × 10−5 214.1
V1 4.170 × 10−5 0.4 0.0 54.79 60.12 4.232 × 10−5 212.8
V2 4.234 × 10−5 0.4 -0.1 54.72 61.00 4.281 × 10−5 204.4
W1 2.042 × 10−5 0.4 0.2 57.89 92.87 2.044 × 10−5 184.6
W2 2.200 × 10−5 0.5 -0.3 57.57 93.43 2.200 × 10−5 169.5
W3 2.139 × 10−5 0.5 -0.5 57.69 92.44 2.139 × 10−5 179.0
W4 2.007 × 10−5 0.5 0.5 57.97 93.22 2.010 × 10−5 186.4
For 5 Maps
K 2.466 × 10−4 0.6 0.8 47.07 22.72 2.519 × 10−4 250.3
Ka 1.442 × 10−4 0.5 0.4 49.40 32.98 1.464 × 10−4 204.5
Q 8.978 × 10−5 0.6 -0.2 51.46 40.66 9.098 × 10−5 216.4
V 4.202 × 10−5 0.4 -0.1 54.76 60.56 4.256 × 10−5 208.6
W 2.097 × 10−5 0.5 0.0 57.78 92.99 2.098 × 10−5 179.3
aSolid angle in azimuthally symmetrized beam.
bRelative error in ΩS .
cRelative change in ΩS between five-year and seven-year analyses.
dForward gain = maximum of gain relative to isotropic, defined as 4π/ΩS . Values of Gm in Table
2 of Hill et al. (2009) were taken from the physical optics model, rather than computed from the
solid angle, and therefore do not obey this relation.
eConversion factor to obtain flux density from the peak WMAP antenna temperature, for a free-
free spectrum with β = −2.1. Uncertainies in these factors are estimated as 0.6, 0.5, 0.6, 0.5 and
0.7% for K, Ka, Q, V and W band DAs respectively.
– 36 –
Table 5. Changes in χ2 from five-year to seven-year beam fits
χ2ν χ2
ν ∆χ2ν
DA (5 year) (7 year)
Side A
all 1.060 1.049 -0.011
K1 1.009 1.006 -0.004
Ka1 1.012 1.016 0.004
Q1 1.078 1.074 -0.003
Q2 1.094 1.102 0.008
V1 1.144 1.155 0.011
V2 1.162 1.171 0.008
W1 1.225 1.171 -0.053
W2 1.239 1.151 -0.088
W3 1.244 1.181 -0.063
W4 1.208 1.156 -0.052
Side B
all 1.065 1.047 -0.018
K1 1.022 1.017 -0.005
Ka1 1.010 1.009 -0.000
Q1 1.093 1.046 -0.046
Q2 1.045 1.052 0.007
V1 1.288 1.171 -0.117
V2 1.178 1.159 -0.019
W1 1.226 1.197 -0.029
W2 1.169 1.155 -0.013
W3 1.223 1.203 -0.021
W4 1.226 1.165 -0.061
– 37 –
Table 6. WMAP Seven-Year CMB Dipole Parameters
dxa dy dz db l b
(mK) (mK) (mK) (mK) () ()
−0.233 ± 0.005 −2.222 ± 0.004 2.504 ± 0.003 3.355 ± 0.008 263.99 ± 0.14 48.26 ± 0.03
Note. — The measured values of the CMB dipole signal. These values are unchanged
from the five-year values and are reproduced here for completeness.
aCartesian components are give in Galactic coordinates. The listed uncertainties include
the effects of noise, masking and residual foreground contamination. The 0.2% absolute
calibration uncertainty should be added to these values in quadrature.
bThe spherical components of the CMB dipole are given in Galactic coordinates and
already include all uncertainty estimates, including the 0.2% absolute calibration uncertainty.
– 38 –
Table 7. Seven-year Spectrum χ2 W-band Null Tests
Data Combinations χ2EE
PTEEE χ2BB
PTEBB
For DAs W1, W2, W3 and W4
yr1 - y2, y3, y4, y5, y6, y7 0.944 0.56 0.976 0.50
yr2 - y1, y3, y4, y5, y6, y7 0.800 0.78 0.956 0.54
yr3 - y1, y2, y4, y5, y6, y7 0.934 0.57 1.166 0.24
yr4 - y1, y2, y3, y5, y6, y7 0.850 0.70 0.920 0.59
yr5 - y1, y2, y3, y4, y6, y7 0.737 0.85 1.286 0.13
yr6 - y1, y2, y3, y4, y5, y7 0.769 0.82 1.101 0.32
yr7 - y1, y2, y3, y4, y5, y6 1.108 0.31 0.831 0.73
yr1, yr2, y3 - y4, y5, y6, y7 0.608 0.96 0.98 0.49
yr1, yr3, y5, y7 - y2, y4, y6 0.860 0.69 1.06 0.38
For 7 years of data
W1 - W2, W3, W4 1.195 0.21 0.982 0.49
W2 - W1, W3, W4 0.802 0.77 0.926 0.58
W3 - W1, W2, W4 0.890 0.64 0.897 0.63
W4 - W1, W2, W3 1.163 0.24 2.061 0.0005
W1, W2 - W3, W4 0.867 0.68 1.361 0.09
W1, W3 - W2, W4 0.866 0.68 1.219 0.19
Note. — χ2 tests for various null combinations of low-ℓ Master polarization power spectra obtained from
uncleaned W-band sky maps. Reduced χ2 values are presented along with the probability to exceed (PTE)
values based on 31 degrees of freedom, corresponding to 2 ≤ ℓ ≤ 32. Polarization cross power spectra were
obtained for each DA year × DA year combination, then appropriately weighted combinations generated
to null any sky signal. Predicted uncertainties were obtained using the standard Fisher matrix formalism
incorporating the inverse noise covariance matrices and the measured sky map noise levels. Since individual
power spectra estimates do include signals (mostly foreground), the uncertainties include a contribution for
the signal × noise cross term as explained in the text. The only anomalous point occurs when the seven-year
W4 data is compared to the seven-year data from the remaining W-band DAs.
– 39 –
Table 8. WMAP Seven-year Cosmological Parameter Summary
Description Symbol WMAP -only WMAP +BAO+H0
Parameters for Standard ΛCDM Model a
Age of universe t0 13.75 ± 0.13 Gyr 13.75 ± 0.11 Gyr
Hubble constant H0 71.0± 2.5 km/s/Mpc 70.4+1.3−1.4 km/s/Mpc
Baryon density Ωb 0.0449 ± 0.0028 0.0456 ± 0.0016
Physical baryon density Ωbh2 0.02258+0.00057
−0.00056 0.02260 ± 0.00053
Dark matter density Ωc 0.222 ± 0.026 0.227 ± 0.014
Physical dark matter density Ωch2 0.1109 ± 0.0056 0.1123 ± 0.0035
Dark energy density ΩΛ 0.734 ± 0.029 0.728+0.015−0.016
Curvature fluctuation amplitude, k0 = 0.002 Mpc−1 b ∆2R
(2.43± 0.11)× 10−9 (2.441+0.088−0.092) × 10−9
Fluctuation amplitude at 8h−1 Mpc σ8 0.801 ± 0.030 0.809 ± 0.024
Scalar spectral index ns 0.963 ± 0.014 0.963 ± 0.012
Redshift of matter-radiation equality zeq 3196+134−133 3232 ± 87
Angular diameter distance to matter-radiation eq.c dA(zeq) 14281+158−161 Mpc 14238+128
−129 Mpc
Redshift of decoupling z∗ 1090.79+0.94−0.92 1090.89+0.68
−0.69
Age at decoupling t∗ 379164+5187−5243 yr 377730+3205
−3200 yr
Angular diameter distance to decoupling c,d dA(z∗) 14116+160−163 Mpc 14073+129
−130 Mpc
Sound horizon at decoupling d rs(z∗) 146.6+1.5−1.6 Mpc 146.2 ± 1.1 Mpc
Acoustic scale at decoupling d lA(z∗) 302.44 ± 0.80 302.40 ± 0.73
Reionization optical depth τ 0.088 ± 0.015 0.087 ± 0.014
Redshift of reionization zreion 10.5± 1.2 10.4± 1.2
Parameters for Extended Models e
Total density f Ωtot 1.080+0.093−0.071 1.0023+0.0056
−0.0054
Equation of state g w −1.12+0.42−0.43 -0.980 ± 0.053
Tensor to scalar ratio, k0 = 0.002 Mpc−1 b,h r < 0.36 (95% CL) < 0.24 (95% CL)
Running of spectral index, k0 = 0.002 Mpc−1 b,i dns/d ln k −0.034± 0.026 −0.022± 0.020
Neutrino density j Ωνh2 < 0.014 (95% CL) < 0.0062 (95% CL)
Neutrino mass jP
mν < 1.3 eV (95% CL) < 0.58 eV (95% CL)
Number of light neutrino families k Neff > 2.7 (95% CL) 4.34+0.86−0.88
aThe parameters reported in the first section assume the 6 parameter flat ΛCDM model, first using WMAP data
only (Larson et al. 2010), then using WMAP +BAO+H0 data (Komatsu et al. 2010). The H0 data consists of a
Gaussian prior on the present-day value of the Hubble constant, H0 = 74.2 ± 3.6 km s−1 Mpc−1(Riess et al. 2009),
while the BAO priors on the distance ratio rs(zd)/DV(z) at z = 0.2, 0.3 are obtained from the Sloan Digital Sky
Survey Data Release 7 (Percival et al. 2009). Uncertainties are 68% CL unless otherwise noted.
bk = 0.002 Mpc−1 ←→ leff ≈ 30.
– 40 –
cComoving angular diameter distance.
dlA(z∗) ≡ π dA(z∗) rs(z∗)−1.
eThe parameters reported in the second section place limits on deviations from the ΛCDM model, first using WMAP
data only (Larson et al. 2010), then using WMAP +BAO+H0 data (Komatsu et al. 2010), except as noted otherwise.
A complete listing of all parameter values and uncertainties for each of the extended models studied is available on
LAMBDA.
fAllows non-zero curvature, Ωk 6= 0.
gAllows w 6= −1, but assumes w is constant and Ωk = 0. The value in the last column is obtained from a combination
of WMAP +BAO data and luminosity distance information obtained from the “constitution” SNe data set (Hicken
et al. 2009) using the methodology described in Komatsu et al. (2010)
hAllows tensors modes but no running in scalar spectral index.
iAllows running in scalar spectral index but no tensor modes.
jAllows a massive neutrino component, Ων 6= 0.
kAllows Neff number of relativistic species, with the prior 0 < Neff < 10.
– 41 –
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